The following approach to the study of geodesic ellipses and hyperbolas is instructive, and leads to a generalisation in Riemannian geometry.

I. In a note in the Mathematical Gazette, February 1934, pp. 43–46, I gave a method for determining the foci, etc., of a conic in general and showed that the method could be applied to the parabola. A direct solution for the parabola can, however, be put down more quickly.

Voici quelques compléments au remarquable article de M. N. M. Gibbins sur la figure constituée par un triangle bordé de triangles équilatéraux.

1. Considérons un triangle BAG, et les triangles équilatéraux BCA1BC1A, AB1C et BA3C, BAC3 , ACB3 construits extérieurement puis intérieurement sur les côtés CB, BA, AC.

There are two main roots to the tree of modern statistical method : the longer one takes us back to the mathematical investigations into games of chance of the seventeenth and eighteenth century mathematicians, the other starts with the statistical researches of Quetelet and Galton.

Of the former we are told by Poisson “that a problem, relative to games of chance, proposed to an austere Jansenist by a man of the world, was the origin of the calculus of probabilities”.

The main object of this article is to suggest that John Napier took the word “ logarithm ”, though not the idea, directly from the Psammites (Arenarius, Sand-Reckoner) of Archimedes. Incidentally it will contain the results of new research into Napier's life, together with an examination of the history of some of the texts of the editio princeps of Archimedes (Basel, 1544) which exist at present in this country.

Though vector analysis is most successful in making the elements of a subject attractive, many problems are solved either cumbrously or by a more or less disguised scalar analysis. Rectangular unit vectors i, j, k are often, and very properly, introduced at an early stage, so that even elementary work may look like Cartesian analysis translated into vector notation. Mathematicians are apt to look askance at oblique axes, since they fall between two stools, having neither the simplicity of the rectangular system nor the adaptability and wide projective outlook of homogeneous coordinates.

Through any point 0 in space a straight line POP1 can be drawn meeting the opposite edges AD, BC of a tetrahedron ABCD in the points P, P 1 respectively; another straight line QOQ1 can be drawn to meet DB, AC in Q, QI; and another straight line ROR1 to meet DC, AB in R, RI.

Students find partial differential equations difficult not only on account of the inherent difficulties of the subject, but because of confusion, omissions, and, frequently, errors in the textbooks. To take an illustration, Piaggio (p. 147, new edition), begins with the statement that the equations

1

2

are equivalent because they represent the same surfaces.

Consider x : y : 1 = t2 - 2t : t : t2 - t - 1. ........................ ( i)

The curve meets the line lx +my + n = 0 in points given by

l(t2 -2t) +mt +n(t2 -t-1) =0.

For any given Z, m, n there are two values of t. Hence the curve meets any line in two points and is therefore a conic. For a given l, m, n the values of t are the roots of the equation

t2 (l +n) + t(m -2l -n) -n =0.

If a number N (other than an odd multiple of 2) is the product of two factors, then N = x2 – y2, where x and y are half the sum and half the difference respectively of the factors, and are integers. To determine x and y we may use the quadratic residues of small primes as follows.

Let N = a2 + b where a2 is the perfect square nearest to N.

The following notes are the outcome of a course of lectures to pupils on invariants. The methods are new to me, but I publish them with hesitation and apologise beforehand should they prove well known.

The following treatment is perhaps easier than, and is in son respects an expansion of, that given by Professor H. F. Baker in h Principles of Geometry, vol. ii, pp. 58-60.

1. A,B,C,O are four given points ; AO meets BG in D, BO meet CA in E, CO meets AB in F. Then DEF is the diagonal point triangle of the quadrangle ABCO. Take this as triangle of reference for areal coordinates and let O be (u, v, w). Then A is (-u, v, w) B is (u, - v, w) and C is (u, v, -w).

By a simplex is meant the figure in general flat space of which the triangle and tetrahedron are examples. In space of n dimensions the simplex has (n + 1) vertices and (n + 1) faces, each face consisting of a prime, i.e. a space of (n - 1) dimensions, containing n of the vertices and hence itself a simplex in (n - 1)-dimensional space. There is a general likeness between the triangle and simplexes of higher order, but this is marred by the fact that, in general, a simplex does not possess an orthocentre.

The study of line geometry by analytical methods involving the use of tangential coordinates and tangential equations is often found difficult by the junior Honours student. In the writer’s view the reason for this is to be found in the fact that, according to the course followed by a good many textbooks, the student finds himself, so to speak, pitchforked into the middle of the subject without any just appreciation of its bearing, or indeed of the need for such a study at all. Some modern writers on analytical geometry have attempted to remedy this by adopting the view that line geometry should be studied from the outset pari passu with point geometry. Unfortunately, however, experience tends to show that this is a mistake, inasmuch as it creates confusion in minds which have had insufficient practice in, and consequently have obtained an inadequate grasp of, analytical methods, which are never at any time found easy of assimilation by the student of elementary mathematics.